Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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In a commutative ring, consists of finite sums , and
Statement
In a commutative ring, consists of finite sums , and .
The empty sum is included and equals .
Facts & Assumptions
Given: A commutative ring and a subset .
is the intersection of ideals containing (The ideal generated by a subset and principal ideals).
The ideal criterion uses subtraction and absorption (Ideal criteria and intersections of ideals).
Multiplication in a commutative ring commutes (Commutative ring).
Proof
Let be the finite sums ; it contains and is closed under subtraction and multiplication by arbitrary ring elements.
Thus is an ideal containing , while every ideal containing contains each such finite sum.
Hence ; taking gives .
Depends on
Used by
- An algebra that is finite dimensional as a vector space over a field is a Noetherian ring Example
- Fields and ℤ are Noetherian, and so are their polynomial rings in finitely many variables Example
- k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
- The polynomial ring in countably many variables is not Noetherian Example
- The subalgebra k[x,xy,xy²,…] of k[x,y] is not Noetherian Example
- The subring k[x,y,x/y,x/y²,…] of k(x,y) has a strictly ascending chain of principal ideals Example
- Working the Hilbert basis construction on an ideal of ℤ[x] with non-monic stages Example
- False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
- A classical zero locus depends only on the generated ideal and its radical Lemma
- A single cancellation step lowers the degree of a polynomial in an ideal once its leading coefficient lies in a realised stage Lemma
- A subring that admits a module retraction from a Noetherian ring is Noetherian Lemma
- An ideal maximal among the non-finitely-generated ideals is prime Lemma
- Every ideal of a localisation is generated by the images of any generating set of its contraction Lemma
- If some ideal is not finitely generated, there is one maximal among the ideals that are not Lemma
- Over a Noetherian ring, an ideal of R[x] is generated by finitely many polynomials realising generators of its stages up to the stabilisation degree Lemma
- The submodule generated by a subset consists of the finite R-linear combinations of that subset Lemma
- A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member Theorem
- A module-finite algebra over a Noetherian ring is a Noetherian ring, and so is every ring between the two Theorem
- Chinese remainder theorem for pairwise comaximal ideals Theorem
- Every quotient and every localisation of a Noetherian ring is Noetherian Theorem
- Regular functions on a principal open are the principal localization Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)